The quasiclassical Keldysh Green function

Transcription

1 The quasiclassical Keldysh Green function Lectures delivered at the Università di Camerino Roberto Raimondi Dipartimento di Fisica, Università di Roma Tre May 2009 c Questa opera è pubblicata sotto una Licenza Creative Commons.

2 Introductory remarks The Keldysh technique is the way to a quantum kinetic equation, but the resulting equations are in general very complicated and difficult to solve The quasiclassical approximation, which works well for degenarate Fermi systems, is a concrete example, where, starting from a quantum formulation, one makes a link to the phenomenologicl Boltzmann equation for quasiparticles The cases which I include in these lectures show that is not only possible to give a microscopic justification to the Boltzmann equation, but also to derive systematically correction terms The quasiclassical approximation, furthermore, even though does not have the elegance of the functional integral methods, provides very often a much clearer physical picture Even at equilibrium, when diagrammatic methods are sufficient, the quasiclassical approximation has the advantage of a more compact derivation I hope to convince you! Let us get started.

3 References 1. L.V. Keldysh, Sov. Phys. JETP 20, 1018 (1965). Where all began 2. L. P. Pitaevskii, E.M. Lifshitz, Physical Kinetics: Volume 10. I find this relatively short account of the basics of the Keldysh technique one of the best. You just learn exactly what is necessary to know in pure Landau style. 3. J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986). This is a classic reference full of hystorical and technical information. It gives the rules of the game 4. P. Schwab and R. Raimondi, Annalen der Physik 12, 471 (2003). This deals mainly with quantum corrections. 5. A. Kamenev and A. Levchenko, ArXiv: v2 (Les Houches). A modern and self-contained account based on functional integrals. The authors really know the subject.

4 Program of the lectures 1. The Keldysh technique 2. The quasiclassical approximation 3. Weak localization 4. Quantum interaction correction 5. Thermal transport and Coulomb Blockade 6. Superconductivity and Andreev scattering

5 1. The Keldysh technique 1. The Keldysh time path 2. The example of the Fermi gas 3. Dyson equation 4. Perturbation theory 5. Disorder and self-consistent Born approximation

6 Let us recall the standard perturbation theory Split the Hamiltonian into free and interacting parts H = H 0 + H I Go to the interaction picture, which in terms of Schrödinger and Heisenberg pictures reads ψ i (t) = e ih 0t ψ S (t) ψ i (t) = e ih 0t e iht ψ H (t) Note that at t = 0 all pictures coincide The time evolution in the interaction picture is given by the S-matrix ψ i (t 1 ) = S(t 1, t 2 )ψ i (t 2 ) The S-matrix is naturally time ordered S(t 1, t 2 ) = T exp i Z t1 t 2 «dt H Ii (t) The problem is how to connect the time evolution of the S-matrix in the interaction picture with the relation between the free and interacting states of the Heisenberg picture

7 Trick of the adiabatic turning on In the far past and future there is no interaction H I e ɛ t H I Then the interaction picture state vector in the far past and future and at t = 0 is related to the free and interacting Heisenberg state vector ψ i (± ) = ψ H0 ψ i (0) = ψ H To have a connection at all times one needs the S-matrix ψ H = S(0, )ψ H0 ψ i (t) = S(t, 0)ψ H From state vectors to operators A H(t) = S 1 (t, 0)A i (t)s(t, 0) By introducing S S(, ) the key formula is ψ H T (A H(t 1 )B H(t 2 )) ψ H = ψ H0 S 1 T (A i (t 1 )B i (t 2 )S) ψ H0 i.e. we have transformed average over the interacting state in an average over the free one

8 The Keldysh Time Contour The problem with the previous formula is that S is time ordered, while S 1 is anti-time ordered The problem is solved by invoking the adiabatic theorem. The state at t =, adiabatically evolved from the state at t = acquires just a divergent phase factor (Gell-Mann and Low (1951)) ψ H0 = exp(iα)ψ H0 S = exp(iα) The unpleasent S 1 factor can be moved into the denominator ψ H T (A H (t 1 )B H (t 2 )) ψ H = ψ H 0 T (A i (t 1 )B i (t 2 )S) ψ H0 ψ H0 S ψ H0 There is an alternative strategy: the Keldysh contour C K = (, ) S (, ), which includes both time-ordered and antitime-ordered factors via the ordering operator T K. The S-matrix becomes Z! S K = T K exp i dt H Ii (t) C K The key formula for perturbation theory becomes ψ H T (A H (t 1 )B H (t 2 )) ψ H = ψ H0 T K (A i (t 1 )B i (t 2 )S K ) ψ H0 For a stationary state both strategies can be used. For a non-equilibrium state one must use the Keldysh strategy.

9 The Keldysh Green function The beauty of the Keldysh time path is that the definition of the Green function is like the equilibrium case G(x 1, x 2 ) = i T K Ψ(x 1 )Ψ (x 2 ), x i = (x i, t i ) The price to pay is that there are actually several Green functions to calculate. It is convenient to introduce a matrix structure in the so-called Keldysh space to indicate which Keldysh-contour branch the time arguments belong to G(x 1, x 2 ) Ǧ = G11 G 12 G 21 G 22 «Time ordered G 11 (x 1, x 2 ) = i T Ψ(x 1 )Ψ (x 2 ) Lesser(Densitymatrix) G 12 (x 1, x 2 ) = +i Ψ (x 2 )Ψ(x 1 ) Greater G 21 (x 1, x 2 ) = i Ψ(x 1 )Ψ (x 2 ) Antitime ordered G 22 (x 1, x 2 ) = i T Ψ(x 1 )Ψ (x 2 ) The Green functions are not all independent and obey the causality condition G 11 + G 22 = G 12 + G 21

10 The Keldysh rotation Due to the fact that the components of Ǧ are not independent, it is convenient to transform the matrix Ǧ as Ǧ « «1 1 Ǧ 1 1 «In this new representation the Green s function is «G R G K Ǧ = G Z G A = 1 G11 G 22 G 12 + G 21 G 11 + G 22 + G 12 + G 21 2 G 11 + G 22 G 12 G 21 G 11 G 22 + G 12 G 21 «The rotated components are G R (x 1, x 2 ) = iθ(t 1 t 2 ) Ψ(x 1 )Ψ (x 2 ) + Ψ (x 2 )Ψ(x 1 ) G A (x 1, x 2 ) = +iθ(t 2 t 1 ) Ψ(x 1 )Ψ (x 2 ) + Ψ (x 2 )Ψ(x 1 ) G K (x 1, x 2 ) = i Ψ(x 1 )Ψ (x 2 ) Ψ (x 2 )Ψ(x 1 ) G Z (x 1, x 2 ) = 0 The vanishing of G Z is not automatic if the evolution on the two time paths is not the same. This may happen in the presence of quantum fluctuations. We will come back to this when treating interactions.

11 The meaning of the rotation The rotation defines classical and quantum fields ψ cl = 1 2 (ψ 1 + ψ 2 ) ψ qu = 1 (ψ 1 ψ 2 ) 2 The connection with the various Green functions G K ψ cl ψ cl G R ψ cl ψ qu G A ψ qu ψ cl G Z ψ qu ψ qu

12 A first example: the Fermi gas at equilibrium One can compute explicitly all the Green functions starting from ψ(x, t) = 1 X e ip x e iξpt V p ξ p = p2 2m µ The retarded and advanced Green function provide information about the energy spectrum G R,A (p, ɛ) = ˆɛ ξ p ± i0 + 1, The Keldysh Green function tells about the distribution function G K (p, ɛ) = G R (p, ɛ)f(ɛ) F(ɛ)G A ɛ (p, ɛ), F (ɛ) = tanh 2T This is basically the the fluctuation-dissipation theorem Observables are given in terms of the lesser Green function G 12 = 1 2 (GK G R + G A ) = f (ɛ)(g R G A ) where f (ɛ) is the Fermi function

13 Dyson equations From the definition, the derivation of the Dyson equation is like in the equilibrium case. For future use, we show both left- and right-side equation of motion (Ǧ 1 0 ˇΣ)Ǧ = δ(x 1 x 2 ) Ǧ(Ǧ 1 0 ˇΣ) = δ(x 1 x 2 ) The differential operator is a matrix in Keldysh space» Ǧ 1 0 (x 1, x 2 ) = i t1 1 2m ( i x + ea(x 1 1)) 2 + eφ(x 1 ) + µ ˇ1δ(x 1 x 2 ) While the electromagnetic potentials φ(x), A(x) are shown explicitly (with e > 0), disorder scattering and electron-electron interactions are contained in the self-energy The self-energy has a triangular structure as the Green function «Σ R Σ ˇΣ K = 0 Σ A The Keldysh Green function can always be expressed in terms of the retarded and advenced Green functions and Keldysh self-energy h (G R,A 0 ) 1 Σ R,Ai G R,A = 1 G K = G R Σ K G A

14 Perturbation expansion in an external potential Let U be one-body external potential. The zero and first order terms of the perturbation expansion are ««««G R Ǧ = 0 G0 K G R 0 G0 A + 0 G0 K U 0 G R 0 G 0 G0 A 0 K 0 U 0 G0 A +... Replace G K 0 with the equilibrium form G K 0 = G R 0 F 0 F 0 G A 0 G K = (G R 0 + G R 0 UG R 0 )F 0 G R 0 (UF 0 F 0 U)G A 0 F 0 (G A 0 + G A 0 UG A 0 ) +... By observing the partial resummations G R = G R 0 +G R 0 UG R , G A = G A 0 +G A 0 UG A , F = F 0 [U, F 0 ]+... One obtains that the equilibrium relation of the Fermi gas is valid in general G K = G R F FG A

15 The standard model of disorder We consider in the Hamiltonian a term Z dxψ (x)u(x)ψ(x) = X p,p c pu(p p )c p The lowest order Born approximation corresponds to the self-energy The average over disorder realizations is indicated by a dashed line and defines the type of disorder model The white noise model U(x)U(x ) = u 2 0δ(x x ) where u 0 is the scattering amplitude

16 Self-consistent Born approximation Let us evaluate the lowest order self-energy Σ R = δ(x x )u 2 0 X p Z dɛ e iɛ(t t ) 2π ɛ ξ p + i0 + Z iδ(x x )2πu0N 2 0 dξ pe i(ξp i0+ )(t t ) µ = i δ(x x ) δ(t t ) τ 1 = 2πu0N 2 0 2τ Actually a self-consistent solution is obtained since what matters is the position of the pole of the Green function in the complex plane, but not its distance from the real axis Σ R = i δ(x x ) 2πN 0 τ X p e i(ξp i/2τ)(t t ) The important physical assumption is that the distance of the pole from the real axis is small compared to the Fermi energy ɛ F τ 1 or p F l 1

17 The form of the self-energy The retarded Green function has the form G R = (ɛ ξ p + i/2τ) 1 G A = (G R ) The self-consistent self-energy can be generalized in the Keldysh space ˇΣ = δ(x x ) Ǧ(x, t; x, t ) 2πN 0 τ The above self-energy can be inserted into the Dyson equation. This form treats the scattering from different impurities incoherently. We will come back to this.

18 Summary of the first lecture In this lecture we have seen how the perturbation theory based on the Keldysh technique works In particular, we have studied the important case of the perturbation expansion for an external potential We have seen how the impurity technique leads to a functional form for the self-energy In the next lecture, we will introduce the so-called quasiclassical approximation, which will allow us to derive effective kinetic equations

19 2. The Quasiclassical approximation 1. The Eilenberger equation 2. Meaning of ξ-integration 3. Observables 4. Gauge invariance and Drude formula

20 The quasiclassical approximation The idea is that in order to describe the response to an external disturbance, it is necessary to keep all the information at the microscopic level The basic assumption is that external disturbances have length scales much larger than the Fermi wave length The arguments of the Green function can be divided in center-of-mass and relative coordinates x = x 1 + x 2 2, r = x 1 x 2, t = t 1 + t 2, η = t 1 t 2 2 The Wigner form corresponds to Fourier transform with respect to the relative coordinates Z G(p, η; x, t) = dre ip r G x + r 2, t + η 2 ; x r 2, t η 2

21 The gradient expansion Suppose to consider a convolution (1 x 1 and so for 2 and 3) Z (A B)(1, 2) = d3 A(1, 3)B(3, 2) Z «« d3 A 2, 1 3 B 2, 3 2 Z = d3 A « , 1 3 B «2 2, 3 2 Z «« d3 A 2, 1 3 B 2, By making the Fourier transform with respect to the relative coordinate of the free arguments 1 2 (A B) p = A(x, p)b(x, p) +... All other terms can be obtained by a gradient expansion (AB)(p, x) = e i( A x B p A p B x )/2 A(p, x)b(p, x)

22 Quasiclassical equation of motion (G. Eilenberger, Z. Phys. 214, 195 (1968) ) Subtract LH and RH Dyson equations from one another so that the delta functions cancel. Then Fourier transform and keep the lowest order term in the gradient expansion»i t + i 1m p x Ǧ(p, x) = hˇσ(p, x), Ǧ(p, x)i The electromagnetic potential enter through the covariant derivatives t Ǧ = [ t ieφ(x, t + η/2) + ieφ(x, t η/2)] Ǧ(p, η; x, t) x Ǧ = [ x + iea(x, t + η/2) iea(x, t η/2)] Ǧ(p, η; x, t) Define the quasiclassical Green function ǧ(ˆp, η; x, t) = i Z π dξǧ(p, η; x, t), ξ = p2 /2m µ The Eilenberger equation h t + v F ˆp i x ǧ(ˆp, η, x, t) = i ˆˇΣ(ˆp, x), ǧ(ˆp, η, x, t), ǧǧ = ˇ1 The normalization condition is necessary since the equation is homogeneous

23 Eilenberger decomposition

24 Meaning of the ξ-integration The momentum integration can be divided into an energy integration, which corresponds to the ξ-integration and an integration over the Fermi surface G R (x 1, x 2 ) = X e ip (x 1 x 2 ) ɛ ξ + i0, r = x + 1 x 2 p Z dp dp e ip r = (2π) 2 ɛ v F (p p F ) p2 2m + i0 + = i ei(p F +ɛ/v F )r Z dp 2 r/2p F = v F 2π e ip s 2πi m p F r 2π ei(p F +ɛ/v F )r = G R 0 (r, ɛ = 0)g R (x 1, x 2 ) At large r = x 1 x 2 integral dominated by the extrema in the exponential and the Green function is factorized in rapidly and slowly varying terms

25 Shelankov analysis (A. L. Shelankov, J. Low Temp. Phys. 60, 29 (1985)) The definition of the quasiclassical Green function appears to be defined by a non-convergent integral The slowly varying part is, however, well defined due to the exponential g R (x 1, x 2 ) = i Z 2π e iξr/v F dξ ɛ ξ + i0 = + eiɛr/v F According to Shelankov, in general, we may write g R (x 1, x 2 ) = i Z dξe iξr/v F G R (p, x), p = pˆr 2π This leads to a well-defined quasiclassical Green function Z «g R i ξr ( ˆp, η; x, t) = lim dξ cos G R (p, η; x, t) r 0 π In the Fermi gas, for instance, v F g R ( ˆp, η; x, t) = δ(η) = g A ( ˆp, η; x, t)

26 Connection to observables and quasiclassical Green function The observables are obtained by evaluating the Green function at equal-time and for doing so it is important to integrate energy first and momentum after ρ(x, t) = ( e)( i) X p Z dɛ 2π G 12(p, ɛ) The momentum integration is divided into ξ-integration and angle integration X Z Z = dξ...,... = d ˆp... p The quasiclassical Green function at equal-time interchanges the correct order of integration g 12 (ˆp, η = 0; x, t) = Z dɛ 2π g 12(ˆp, ɛ; x, t) Already, at equilibrium, for the Fermi gas case, this leads to problems since g 12 (ˆp, ɛ; x, t) = 2f (ɛ), f Fermi function which is not integrable

27 Continuity equation At equilibrium the Keldysh component is well defined since g K (ˆp, ɛ; x, t) = g R (ˆp, ɛ; x, t)f 0 (ɛ) F 0 (ɛ)g R (ˆp, ɛ; x, t) = 2(1 2f (ɛ)) is integrable due to antisymmetry of the hyperbolic tangent and g K (ˆp, η = 0; x, t) = 0 The Eilenberger equation for g K with φ = 0, A = 0 and Σ = 0 reads ( t + v F ˆp x)g K = 0 Then by averaging over ˆp and integrating over ɛ t en 0 2 Z dɛ g K + x en 0 2 Z dɛ v F ˆpg K = 0 which yields the continuity equation for charge and current δρ(x, t) = en 0 2 Z dɛ g K (ˆp, ɛ; x, t) j = en 0 2 Z dɛ v F ˆpg K

28 Continuity equation in the presence of potentials Let us insert the potentials φ and A ( t e tφ(x, t) ɛ + v F ˆp x + ev F ˆp ta(x, t) ɛ)g K = 0 Again by averaging over ˆp and integrating over ɛ tδρ(x, t) + x j(x, t) = 0 with since δρ(x, t) = en 0 2 j(x, t) = en 0 2 Z Z dɛ g K (ˆp, ɛ; x, t) 2e 2 N 0 φ(x, t) dɛ v F ˆpg K Z dɛ ɛg K ( ˆp, ɛ; x, t) = g K ( ˆp, ɛ; x, t) = g K eq( ˆp, ɛ; x, t) = 4 The expression for the density shows clearly how the quassiclassical Green function captures the low energy part

29 Impurity scattering and kinetic equation Disorder in the Born approximation just shifts the pole in the complex plane for the retarded and advanced Green functions, which then do not change g R = g A = 1 The RHS of the Keldysh component of the Eilenberger equation yields [ˇΣ, ǧ] K = Σ R g K + Σ K g A g R Σ K g K Σ A = i τ gk 2Σ K = i 1 X τ gk 2 G K (p, ɛ; x, t) 2πN 0 τ = i τ (gk g K ) which has the form of the collision integral with the scattering-in and scattering-out terms well-konwn in the Boltzmann s equation within the relaxation-time approximation p

30 Drude conductivity and gauge invariance: vector gauge A = te ( t + v F ˆp x ev F ˆp E ɛ)g K = 1 τ ( gk g K ) By looking for a uniform solution, we take the p-wave component of both sides ˆpg K = ev F τ d E ɛgk To linear order in the electric field By integration over the energy ɛ ˆpg K = ev F τ d E ɛgk eq j = en 0v F ˆpg K = en 0v F 2 2 4ev F τ E = σ D E 2

31 Drude conductivity and gauge invariance: scalar gauge φ(x) = E x Apparently the scalar potential drops out ( t + v F ˆp x)g K = 1 τ ( gk g K ) However ˆpg K = v F τ 2 x gk We must use the condition of charge uniformity xδρ(x, t) = x en0 2 Z «dɛ g K (ˆp, ɛ; x, t) 2e 2 N 0 φ(x, t) = 0 The bottom line is the minimal substitution also in this case x x ee ɛ

32 Summary of the second lecture In this lecture we have derived the Eilenberger equation for the quasiclassical Green function We have applied this equation to derive the semiclassical theory of electrical transport In the next lectures we are going to apply the Eilenberger equation to different situations to obtain corrections to the semiclassical level

33 3. Weak Localization 1. Crossed diagrams 2. Backscattering 3. Corrections to conductivity

34 Some hystorical remarks Anderson localization was invented in 1958, half a century ago and perhaps would deserve an entirely dedicated lecture course Here, I concentrate on the phenomenon of weak localization, which is a perturbative effect on the metallic behavior My emphasis will be on the advantage of the quasiclassical method in order to capture the physical origin of the phenomenon

35 Quantum interference and crossed diagrams The Born approximation corresponds to the Rainbow diagrams. Interference between scattering events at different sites is not allowed The maximally crossed diagrams describe the interference between time-reversed paths (G. Bergmann, Phys. Rep., 107, 1 (1984).)

36 Corrections to the self-energy Let us consider a term for the Keldysh Green function G R UG R... G R UG K UG A... G A UG A Note that an equal number of G R and G A precede and follow G K We take the disorder average selecting the diagrams with maximum crossing δσ K = Disorder average yields momentum conservation at each impurity line Q = p + p p = Q p p 1 = Q p 1 p 2 = Q p 2 p 3 = Q p 3

37 The Cooperon channel The so-called Cooperon is the sum of all diagrams with an arbitrary number of crossed impurity lines (J. S. Langer and T. Neal, Phys. Rev. Lett. 16, 984 (1966).) C(Q, ω) = ɛ + ω 2 ɛ ω 2 The correction to the self-energy is a convolution with respect to the relative momentum δσ K (p, ɛ; q, ω) = X Q C(Q, ω)g K (Q p, ɛ; q, ω) The momentum Q describes the effect of multiple scattering

38 The building block η RA = X p Diffusion pole and backscattering By using the time-reversal invariance one line can be reversed turning the crossed diagrams into a geometric series of ladder diagrams. The ladder is a particle-particle scattering channel as in the Cooper mechanism for superconductivity G R (p, ɛ + ω 2 )GA (Q p, ɛ ω 2 ) = 2πN 0τ h i 1 + iωτ DQ 2 τ +... C(Q, ω) = 1» 1 2πN 0 τ ηra 2πN 0 τ + = 1 1 2πN 0 τ 2 DQ 2 iω 1 The diffusion pole selects processes with p p

39 A technical detail X G R (p, ɛ)g A (p, ɛ) = p Z µ dξ Z N 0 dξ = 2πN 0 τ N(ξ) (ɛ ξ + i/2τ)(ɛ ξ i/2τ) 1 (ɛ ξ + i/2τ)(ɛ ξ i/2τ) All integrals involving a number of retarded and advanced Green function can be made in the same way by using the residue Cauchy theorem. It is important to note that if all the Green functions are retarded or advanced the integral vanishes

40 Quasiclassical procedure The ξ-integration yields the final form to be used in the Eilenberger equation Z 1 h i K Z dξ δ ˇΣ, π Ǧ 1 = dξ δσ R G K + δσ K G A G R δσ K G K δσ A π By using G K = G R F FG A = 1 2 G R g K g K G A Correction to the scattering-in term of the kinetic equation δσ k = 1 Z h dξ G R p, ɛ + ω G A p, ɛ ω i 2 1 π gk ( ˆp, ɛ, q, ω) X C(Q, ω) (S. Hershfield and V. Ambegaokar, Phys. Rev. B 34, 2147 (1986).) But the story is not complete yet, since we dot know about diagrams when the numbers of G R and G K before and after G K is not the same Q

41 The arising of the Hikami box In addition to this diagram, we must consider that obatined by the interchange of R and A Correction to the scattering-out term of the kinetic equation δσ K = 1 2πN 0 τ X p 1 G K (p 1, q, ɛ, ω) X Q p = Q p + q/2 p 1 = Q p 1 + q/2 p 2 = Q p 2 p 3 = Q p 3 G R Q p 1 + q 2, ɛ + ω G R Q p + q 2 2, ɛ + ω 2 C(Q, ω)

42 Kinetic equation with quantum corrections ( t + v F ˆp x)g K (ˆp, η, x, t) = 1 h i g K (ˆp, η, x, t) g K (ˆp, η, x, t) τ Z + 1 dt α(t t ) τ h i g K ( ˆp, η, x, t ) g K (ˆp, η, x, t ) α(t t ) = 2τ 2 X Q Z dω ) 2π e iω(t t C(Q, ω) The correction to the in- and out- terms maintain charge conservation The non local-in-time response in the collision integral corresponds to the memory effect of a self-retracing trajectory

43 Weak localization correction j =» Z 1 dt α(t t ) σ D E α 2D (t t ) = Θ(t t ) 1 (2π) 2 N 0 D t t Renormalization of conductivity In 2D logarithmic corrections Z dt α(t t ) Z t τ dt 1 1 τ φ (2π) 2 N 0 D t t = 1 (2π) 2 N 0 D ln τ φ τ The controlling parameter is the conductance measured in units of the quantum of conductance σ = σ D 2e2 h 1 2π ln τ φ τ

44 Scaling theory and MIT τ time above which diffusive motion set in τ φ maximum time over which phase memory lasts in general τ φ T p in finite system at low temperatures τ φ L 2 /D Dimensionless conductance Effective perturbation theory g = σ DL d 2 2e 2 /h σ = σ D π g ln L «l L. P. Gorkov, A. I. Larkin, and D. E. Khmelnitskii, Pis ma Zh. Eksp. Teor. fiz. 30, 248 (1979) [JETP Lett. 30, 228 (1979)]. E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).

45 Summary of the third lecture In this lecture we have applied the quasiclassical Green function approach to weak localization We have derived corrections term to the kinetic equation associated to the backscattering mechanism originating from the interference of self-retracing trajectories In the next lecture we are going to illustrate a different quantum correction originating from the electron-electron interaction

46 4. Quantum Interaction Corrections 1. Electron-electron interaction 2. Density of states 3. Electrical conductivity

47 Diffusive approximation in general Let us recall the Eilenberger equation for the quasiclassical Green function [ t + v F ˆp x] ǧ = 1 [ ǧ, ǧ], ǧǧ = ˇ1 2τ Diffusive motion makes the Green function almost isotropic and allows expansion in s-wave and p-wave ǧ = g s + ˆp ǧ p +..., ǧ sǧ s = ˇ1, o nǧ s, ǧ p = 0 One obtains the Usadel diffusive equation (K. D. Usadel, Phys. Rev. Lett. 25, 507 (1970).) tǧ s D x ǧ s xǧ s = 0 The generalized current is related to the gradient of the s-wave component ǧ p = v F τǧ s xǧ s

48 How to include electron-electron interaction Hubbard-Stratonovich transformation Interaction mediated by a boson (photon) field φ (G. Zala, B. N. Narozhny, and I. L. Aleiner, Phys. Rev. B 64, (2001)) Boson on the Keldysh contour D(x 1, x 2 ) = i T K φ(x 1 )φ(x 2 ) D 11 (x 1, x 2 ) = i T φ(x 1 )φ(x 2 ) D 12 (x 1, x 2 ) = i φ(x 2 )φ(x 1 ) D 21 (x 1, x 2 ) = i φ(x 1 )φ(x 2 ) D 22 (x 1, x 2 ) = i T φ(x 1 )φ(x 2 ) As in the case of Fermions, also Boson Green functions obey the condition D 11 + D 22 = D 12 + D 21 Boson-fermion interaction matrix structure In Keldysh space «Ψ ˆφΨ Ψ φ φ 2 Ψ φ1 (x) t (, ) φ 2 (x) t (, )

49 By making the Keldysh rotation as for fermions n o D K (x 1, x 2 ) = i φ(x 1 ), φ(x 2 ) D R (x 1, x 2 ) = iθ(t 1 t 2 ) [φ(x 1 ), φ(x 2 )] D A (x 1, x 2 ) = iθ(t 2 t 1 ) [φ(x 1 ), φ(x 2 )] Notice the anticommutator in the Kedysh component and the commutator in the retarded and advanced components By means of the Keldysh rotation Ψ ˆφΨ Ψ ˆφ Ψ «φ 1 0 ˆφ = 0 φ 2 ˆφ = 1 «φ 1 + φ 2 φ 1 φ 2 2 φ 1 φ 2 φ 1 + φ 2 φ 1 σ 0 + φ 2 σ 1 φ 1 classical component φ 2 quantum component T K ˆφ (x 1 ) ˆφ (x 2 ) = i 2 D K (x 1, x 2 ) «D R (x 1, x 2 ) D A (x 1, x 2 ) 0

50 Interaction propagator in a diffusive (Fermi gas) metal As for fermions, the Keldysh component is expressed in terms of the distribution function, which at equilibrium has the standard form of the Bose statistics h i D K (q, ω) = D R (q, ω) D A ω (q, ω) coth 2T The retarded and advanced part are given by the RPA resummation for standard screening D R (q, ω) = V (q) 1 V (q)χ(q, ω) χ(q, ω) = 2N 0 Dq 2 Dq 2 iω I Dq 2 > ω D R (q, ω) 2πe2 κ 2D II Dq 2 < ω < Dqκ 2D D R (q, ω) 2πe 2 iω Dq 2 III Dqκ 2D < ω D R (q, ω) 2πe2 q Various regimes of transferred momentum and frequency I. Physics of diffusive modes and interference effects II. Physics of coupling to modes of the e.m. environment

51 Eilenberger equation with interaction: three steps Expand in powers of φ ǧ = ǧ (0) + ǧ (1) + ǧ (2) +..., ǧ (0) = «1 2F 0 1 Solve up to quadratic order in φ h i tǧ s D x ǧ s xǧ s = i ˆφ, ǧ «φ 1 φ ˆφ 2 = φ 2 φ 1 Average over the fluctuations of φ Notice that for an external (classical) field φ 2 = 0 due to causality. Since for a quantum field φ 2 0, the Green function has an extra component g Z, which, however, must vanish after averaging g R ǧ = g Z g K g A ««g R ǧ φ = φ g K φ 0 g A φ

52 How it actually works in more detail The normalization condition ǧ 2 = ˇ1 makes only two components independent g R = (g(1)z + g (1)K g (1)Z ) +... g A = (g(1)k + g (1)Z g (1)K ) +... Up to second order in φ, g (1)Z and g (1)K are enough, because the current operator (ǧ xǧ) K φ = g R xg K + g K xg A φ has a correction of the form δ(ǧ xǧ) K φ = x g (2)K φ hg (1)K ( xg (1)Z )2F + 2F( xg (1)Z )g (1)K i φ After averaging, the Keldysh component must be related to retarded and advanced components via the distribution function g (2)K φ = g (2)R φ F F g (2)A φ

53 First order equations One obtains non homegenous linear differential equations h i ( t D x)g 2 (1)K (η, x, t) = 2i φ 1 (t + η/2, x) φ 1 (t η/2, x) F(η, t) ( t + D 2 x)g (1)Z (η, x, t) = 2iδ(η)φ 2 (t, x) The solution can be written in terms of the appropriate Green function for the diffusion operator Z g (1)K (η, x, t) = 2i g (1)Z (η, x, t) = 2iδ(η) h dt x L tt (x, x ) φ 1 (t + η 2, x ) φ 1 (t η i 2, x ) Z dt x L t t(x, x )φ 2 (t, x ) F(η, t ) The Green function for the diffusion operator corresponds to the ladder resummation of impurity lines in the diagrammatic approach ( t D 2 x)l tt (x, x ) = δ(t t )δ(x x )

54 δn N 0 = g (2)R (ɛ, x, t) φ = i X q Density of states Z dω D R (q, ω) ɛ ω 2π (Dq 2 iω) tanh 2 2T The diffusion poles come from g (1)Z and g (1)K The distribution function comes from g (1)K The interaction comes from φ 1 φ 2!!" Comparison with diagrammatic approach (B. L. Altshuler, A.G. Aronov A, and P. A. Lee, Phys. Rev. Lett., 44, 1288 (1980). ) $%"& $!!#!#!"!!"!!##!"!##!!#!##

55 Degree of singularity of the corrections anf physical origin In 2D for short range interactions the correction is logarithmic δn(ɛ) N 0 = N 0 V R (0, 0)t ln ɛτ and controlled by the parameter t = 1/(2π) 2 N 0 D 1/g as in the weak localization This is the famous zero-bias anomaly in DOS However for long range interaction the correction is log-square! δn(ɛ) = t «ɛ 4 ln ( ɛ τ) ln τd 2 κ 4 2D This is due to the fact that the integral is dominated by regime II for transferred momentum and frequency Regime II corresponds to long distances where interference effects are no longer effective. We will come back to this point.

56 Electrical Current j = en 0D 2 Z dɛ(ǧ s xǧ s) K = δj Born + δj QC (a) (c) (c) (d) (b) (e) Diagrams (c), (d) and (e) correspond to δj Born and eventually cancel (a) and (b) correspond to δj QC and yield the interaction corrections (B. L. Altshuler, A.G. Aronov A, and P. A. Lee, Phys. Rev. Lett., 44, 1288 (1980). )

57 Structure of the corrections Note that δj Born has the same structure as the density of states gives a term proportional to the gradient of the density and vanishes at uniform density as in a linear response calculation δj Born = D x en 0 2 Z dɛδ g (2)K (ɛ, x, t) φ = D xδρ(x, t) δj QC has three ladders due to the internal derivative the factor Dq 2 and F xf make relevant regime I for the Coulomb interaction yileds the quantum interaction correction to the elctrical conductivity Z Z dω δj QC = N 0 D dɛ X 2π q (F(ɛ, x) xf(ɛ ω, x))2 D2 q 2 D R (q, ω) (Dq 2 iω) 3 «δσ = e2 1 2π 2 ln T τ

58 !"#$%&$%$"'$()&(*'+##$%,"-(&%).(,./0%,#,$*( +"1(*$2&3')"*,*#$"#(/)#$"#,+2 4"($2$'#%)"(.)5$*(+2)"-(/+#6( 4()%(7 86$(/6+*$(1,&&$%$"'$( 9$#:$$"(/+#6* 4(+"1(7(,*(10$( #)(#6$($;#%+('2)*$1(2))/()&( /+#6(7!"#$%&$%$"'$(9$:#$$"( /+#6* 4(+"1(7 4(9+'<-%)0"1($2$'#%)"(-)$*( +%)0"1(#6$( *+.$('2)*$1(2))/(+2)"-(/+#6(=(+"1(,"#$%+'#,"-(:,#6( #6$()#6$%($2$'#%)"('+"'$2*(#6$( $;#%+(/6+*$()&(/+#6(7 >6$").$"+(,"(%$-,.$(! # " " " "!! #!!

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60 Summary of the fourth lecture We have set the formalism for e-e interaction in disordered systems Corrections to density of states and conductivity Identification of two important regimes of effects

61 5. Thermal transport and Coulomb blockade 1. Definition of heat current 2. Quantum corrections and W-F law 3. Transport through a tunnel junction 4. Coulomb Blockade

62 Some history Pre-history Wiedemann-Franz Boltzmann-Drude-Sommerfeld Classical period Chester and Tellung (1960) Langer (1962) κ σt = L 0 = π2 kb 2 3e 2 General validity based on: Independent quasiparticles Fermion statistics Elastic scattering Modern period Castellani, Di Castro, Kotliar, Lee, Strinati (1987,1988) Livamov, Reizer, and Sergeev (1991) Niven and Smith (2003) With quantum corrections

63 Heat current at semiclassical level Consider the energy density in terms of the quasiclassical Green function Z ρ Q = πn 0 ( i η)gs K (η, x, t) η=0 = N 0 dɛ ɛgs K (ɛ, x, t) Check that it gives the correct value for the specific heat of the electron gas by using gs K (η, x, t) = 2i π P 1 η + 2iπ T 2 6 η +... so that ρ Q = 2π2 N 0 T 2 6 Derive each term of the Eilenberger equation with respect to η, the relative time, set η = 0 and take the angle average t( i η)g K s (η, x, t) η=0 + x v F d ( i η)gk p (η, x, t) η=0 = 0 One gets a continuity equation for the energy and an expression for the energy current j Q = πn 0v F ( i η)gp K (η, x, t) η=0 d

64 A fourier transform

65 The semiclassical result The p-wave is related to the s-wave by the usual relation g K p (η, x, t) = l xg K s (η, x, t) Space dependence via local equilibrium T (x) l xgs K (η, x, t) = 2iπl T (x) η xt (x) 3 By taking the derivative with respect to η i η 2iπl T (x) «η xt (x) = ( xt (x)) 2πl T (x) 3 3 The thermal current becomes j Q = ( xt (x)) π2 3 2N 0 v F l T (x) = ( xt (x)) π2 d 3e σ DT 2

66 Quantum Corrections to Heat Current The strategy is as for electrical current: There are two terms δj Q = N 0D 2 Z dɛ ɛδ(ǧ s xǧ s) K = δj a Q + δj b Q Instead of the charge vertex e, there appears an energy vertex ɛ The first term has the same structure as the density of states Z Z δj a dω Q = DN 0 T dɛ ɛ 2π T (F ɛ ω(x)f ɛ(x)) Im X D R (ω, q) ( iω + Dq 2 ) 2 q The second term instead is like the quantum corrections to the electrical conductivity Z Z δj b Q = DN 0 T dɛ ɛ dω 2π Fɛ(x) T F ɛ ω(x) 4 d Im X q Dq 2 D R (ω, q) ( iω + Dq 2 ) 3

67 Conclusions about the Wiedemann-Franz law Previous literature shows some controversy Castellani, Di Castro, Kotliar, Lee, Strinati (1987) RG analysis confirms W-F Livanov, Reizer, Sergev (1991) finds extra terms that violate W-F Niven and Smith (2003) also find extra terms Apparently no explanation for the different results Our analysis clarifies the issue The final result shows an extra term which violates the W-F law κ = π2 kbt 2 σ 3 e 2 D + δσ + 1 «e 2 2 πh ln( Dκ2 2d/k B T ) The term that is in agreement with W-F comes from integration of diffusive modes corresponding to energies T < ω < τ 1 which is regime I The term violating W-F is due to energies ω < T in regime II and is beyond the reach of RG analysis since the singularity is purely infrared

68 The Coulomb blockade theory (Yu. V. Nazarov, Zh. Eksp. Teor. Fiz. 95, 975 (1989) [JETP Lett. 66, 214 (1989)]. ) The physics associated to the violation of the W-F is also clearly seen when considering the Coulomb interaction affecting transport through a tunnel junction Left Right In the Eilenberger equation we neglect space dependence but for which side the quasiclassical Green function belongs to h i tǧ i (η, t) = ie ˆφ i, ǧ i, i = L, R 0 x The solution can be expressed as a gauge transformation, since the fluctuating Hubbard-Stratonovich field depends on time only Z ǧ i (η, t) = e i ˆϕ i (t+η/2) ǧ 0,i (η, t)e i ˆϕ i (t η/2) ˆϕ i (t) = dt ˆφ i (t) If only one electrode were present the field φ could be eliminated by a gauge transformation. This explains why the singularities in the density of states drop out from physical quantities

69 Tunneling current We need boundary conditions connecting the quasiclassical Green functions on the two sides of the junction In general, for arbitrary tunnel transmittance this is a difficult problem We confine to the tunneling limit within the Bardeen model with tunneling amplitude T LR ˇΣ L = T LR Ǧ R (0, 0)T LR ˇΣ R = T RL Ǧ L (0, 0)T RL The tunneling current is then the Keldysh component of the commutator j = G T 8e Z G T T LR T LR tunneling conductance dɛ [ǧ L (ɛ, t), ǧ R (ɛ, t)] K φ In the absence of interaction standard tunneling theory

70 How interaction affects tunneling by changing the density of the states Gauge transformation: eliminate the interaction from one lead, say the left one The average over the fluctuating field factorizes j = G Z «T ɛ + dɛ(gl R gl A )(gr R gr)(f A evl,r R F L ) F L,R = tanh 8e 2T What is left is the average over the φ field of the retarded Green function «gr R (t 1, t 2 ) φ = e i ˆϕ ( t 1) δ(t1 t 2 ) 2F (t 1, t 2 ) e i ˆϕ ( t 1) 1i 0 δ(t 1 t 2 ) φ j1 The matrix structure is associated to the quantum component φ 2 e ±i ˆϕ ( t) = e ±iϕ1 (t) cos ϕ 2 (t) «±i sin ϕ 2 (t) ±i sin ϕ 2 (t) cos ϕ 2 (t) ij e iφ φ = e (1/2) φ2

71 P-theory The effect of the interaction is condensed into a single function Z» J(t) = e 2 dω 1 cos(ωt) 2π ImDR (ω) i sin(ωt) ω 2 ω and in the Fourier transform of its exponential P(ω) = Z dte iωt e J(t) The correction to the density of states depends of the spectral density of the interaction δ g R R (ɛ) φ = Z dω 1 2π 2 (F R(ɛ + ω) F R (ɛ ω))p(ω) G.-L. Ingold and Yu. V. Nazarov, in [91], chapter 2 Single Charge Tunneling, edited by H. Grabert and M. Devoret, NATO ASI Series B, Vol. 294, (Plenum Press, New York, 1992).

72 Resistor-Capacitor Model q C R When an electron tunnels through the junction, a charge is added to the (right) lead The charge creates an extra potential After some time, the charge relaxes and the lead goes back to a charge neutrality condition The process may be represented in terms of an equivalent circuit with a capacitor and a resistor in parallel

73 The solution of the circuit equations where τ = RC Q(t) = qe t/τ U(t) = q C e t/τ Resistor-Capacitor Model: continued allows to obtain the effective interaction tq(t) = qδ(t) U(t) = Z dt D R (t t )Q(t ) which defines the screened interaction By setting q = e and defining the charging energy E c = e 2 /(2C) Z» dω 1 1 cos(ωt) J(t) = E c i sin(ωt) πτ ω 2 + τ 2 ω 2 ω In the limit of large τ and for T 0 J(t) ie ct E ctt 2 P(ω) = 2πδ(ω + E c) Suppression of the density of states and blocking of tunneling Resummation of the zero-bias anomaly in DOS g R R (ɛ) φ = Θ( ɛ E c)

74 Summary of the fifth lecture Thermal transport: continutity equation and heat current Two types of corrections to the heat current Coulomb blockade

75 6. Superconductivity 1. General equation 2. Landau-Ginzburg limit 3. Andreev scattering and ZBA

76 Quasiclassical formulation o Eilenberger s eq. t nˇσ z, ǧ iɛ [ˇσ z, ǧ] D xǧ xǧ = i ˆ ˇ, ǧ ǧǧ = ˇ1 We consider the diffusive limit, but the clean limit can be also analyzed «ˆσz ˆ0 Keldysh space ˇσ z = ˆ0 ˆσ z The phase of the order parameter is fixed ĝr «ĝ K ǧ = ˆ0 ĝ A «ˆ ˆ0 ˇ = ˆ0 ˆ Nambu space ĝ R,A = G R,Aˆσ z + if R,Aˆσ y, ĝ K = ĝ Rˆf ˆf ĝ A ˆf = f0ˆσ 0 + f z ˆσ z The distribution function f 0 ± f z refer to particles and holes, respectively Self-consistency ˆ = gn 0 π 2 ĝk (η = 0, x, t) ˆ = i ˆσy Larkin A I and Ovchinikov Y N in Non equilibrium superconductivity 1986 eds. Langeberg D N and Larkin A I (NorthHolland)

77 Uniform equilibrium case Distribution functions ɛ f 0 = tanh 2T f z = 0 Normalization condition (G R ) 2 (F R ) 2 = 1 Explicit solution G R,A ɛ (ɛ) = ± p F R,A sign(ɛ) (ɛ) = ± p (ɛ + i0+ ) 2 2 (ɛ + i0+ ) 2, 2 = gn 0π 2 Z dɛ ɛ 2π tanh (F R (ɛ) F A (ɛ)) 2T

78 Direct evaluation from BCS theory The same expression can of course be obtained by the ξ-integration of the BCS Green function In BCS we have up 2 = 1 «1 ± ξp vp 2 = 1 up 2 Ep 2 = ξp E p G R = i π = = = Z Z Z 0 d ξ p up 2 ɛ E p + i0 + vp 2 + ɛ + E p + i0 + d ξ p u pδ(ɛ 2 E p) + vpδ(ɛ 2 + E p) d ξ pδ(ɛ E p) ɛ p (ɛ + i0+ ) 2 2!

79 Landau-Ginzburg Equation for the order parameter Consider the Eilenberger equation for the anomalous part F and express G using the normalization condition ( 2iɛ D 2 x)f R,A (ɛ, x) = 2i (x)(1 + (1/2)(F R,A (ɛ, x)) 2 ) At first limit to linear term in and F Z F R,A (ɛ, x) = ± d x Pɛ R,A (x x ) (x ) The Green function P plays the role of propagator of the Cooper pair ( 2iɛ D 2 x)p R,A ɛ (x x ) = 2iδ(x x ) To solve the integral equation, expand the order parameter (x ) = (x +x x) = (x)+(x x) x (x )+ 1 2 (x x) i (x x) j 2 ij (x)+... Plug F into the self-comsistency equation» gn 0 ln T + 7ζ R(3) T c 8π 2 T 2 2 (x) πd x 2 (x) = 0 8T c

80 A technical detail: the coefficient of the linear term Z = (x) = (x) dɛ ɛ 2π tanh 2T Z Z 4 (x) ln ω D T dɛ ɛ 2π tanh 2T dɛ 2π tanh ɛ 2T (F R (ɛ, x) F A (ɛ, x)) Z ( Z d x Pɛ R (x x ) + «1 ɛ + i ɛ i0 + d x P A ɛ (x x ))

81 A technical detail: the coefficient of the derivative term Z x (x) = i D 2 2 x (x) = i D 2 2 x (x) = πd 2T Z Z dɛ ɛ Z 2π tanh ( 2T Z d x Pɛ R (x )x 2 + d x P A ɛ (x )x 2 ) dɛ ɛ «2π tanh 1 2T (ɛ + i0 + ) 1 2 (ɛ i0 + ) 2 dɛ ɛ «2π tanh 1 ɛ 2T (ɛ + i0 + ) 1 (ɛ i0 + )

82 A technical detail: the coefficient of the cubic term Solve the cubic term by using the first order result F R,A(3) (ɛ, x) = ± 1 Z «3 dx Pɛ R,A (x ) 3 (x) = ± 1 3 (x) 2 2 (ɛ ± i0 + ) 3 To Plug it into the self-consistency condition, integrate over energy (x) = 1 π 2 T 2 = 7ζ R(3) 8π 2 T 2 Z X n=0 dɛ ɛ «2π tanh 1 2T (ɛ + i0 + ) (ɛ i0 ) 3 1 (2n + 1) 3

83 Tunneling current The boundary condition in the tunneling limit as in the case of Coulomb blockade (M. Y. Kuprianov and V. F. Lukichev, Sov. Phys. JETP 64, 139(1988).) Ǐ = σ e ǧr xǧ R = G T 2e [ǧ R, ǧ L ], We allow for the phase of the order parameter ĝ R(A) i = (if R(A) i The current has two components j = G T 8e sin(φ i ), if R(A) cos(φ i ), G R(A) Z i dɛ(i J + I PI ) i ) The first term I J is the Josephson current which is different from zero only the order parameter on the two sides of the junction has a different phase h i I J = isin(φ 1 φ 2 ) f 0R (FR R FR A )(FL R + FL A ) + f 0L (FR R + FR A )(FL R FL A ) In the Josephson only the symmetrix part of the distribution function enters f 0L,R = 1 2 (tanh((ɛ + ev L,R)/2T ) + tanh((ɛ ev L,R )/2T )) Clearly the Josephson current exists even at zero voltage drop

84 Andreev scattering A. F. Andreev, Sov. Phys.JETP 19, 1228 (1964). The second term is called quasiparticle and interference current h i I PI = (GL R GL A )(GR R GR) A + cos(φ 1 φ 2 )(FR R + FR A )(FL R + FL A ) (f zl f zr ). It can be non zero only when a bias is applied f zl,r = 1 2 (tanh((ɛ + ev L,R)/2T ) tanh((ɛ ev L,R )/2T )) The terms with G R and G L yields the standard quasiparticle current The terms with F L and F R yields Andreev scattering Andreev scattering and proximity effect These two phenomena are two aspects of the same thing: at the S-N interface one electron is scattered back as a hole and this makes the Gorkov s function F leaking in the normal side, i.e., yielding the proximity effect

85 Andreev scattering in pictures Blonder, Tinkham, Klapwijk (1982), Shelankov (1980)

86 NIS structure Left S I N Right Experiments show an enhancement at low voltage (Kastalskii et al. (1991)) that cannot be explained by the simple BTK theory 0 L j = G Z h i T dɛ (FR R + FR A )(FL R + FL A ) (f zl f zr ). 8e x F R L 0 Superconductor at equilibrium F R R (ɛ, x), f z(ɛ, x) To be determined by solving the kinetic equations

87 Kinetics equations (Zaitsev (1990), Volkov and Klapwijk (1992)) First solve the spectral problem xĝ R(A) xĝ R(A) = iɛ hˆσ z, ĝ R(A)i Plug the result into the equation for the Kedysh component Express ĝ K = ĝ Rˆf ˆf ĝ A x(ĝ R xĝ K + ĝ K xĝ A ) = 0 x h xˆf ĝ R ( xˆf )ĝ A i (ĝ R xĝ R )( xˆf ) ( xˆf )(ĝ A xĝ A ) = 0 Finally obtain a diffusion-like equation with an energy and position-dependent coefficient x h (1 G R G A F R F A ) xf z i = 0 Z fz(l) fz(0) x dx f z(x) = m(x) + f z(0) m(x) = m(l) 0 1 G R (x )G A (x ) F R (x )F A (x )

88 Zero bias anomaly (ZBA) On the normal side the diffusion equation with boundary condition (for L ) F(ɛ, ) = 0 yields F R (ɛ, x) = F F (ɛ, 0)e i 2i ɛ /Dx F R R e i 2i ɛ /Dx To connect the Gorkov s function on the two sides we use the standard boundary conditions (current conservation) which are valid for a weak proximity effect σs e ĝr R R ĝ R R = G T 2e [ĝr R, ĝ R L ] if R R = s D G T 2i ɛ 2σS F L R One sees that the Gorkov function on the normal side is smaller by a factor G T with respect to that on the superconducting side but there is square-root of the energy at the denominator The smallness of the factor G 2 T is compensated by the an effective conductance of the normal side (at low T, L T L) G NIS G2 T G eff N G eff N = σs r D L T = L T T

89 Summary of all the lectures With superconductivity we have ended our tour through the applications of the quasiclassical Green function approach My aim has been that of showing the flexibility of the method and its vaste range of applicability I think that in most cases is more powerful of the conventional diagrammatics, but its application is not straightforward. It requires some physical understanding of the physical phenomenon under study A current field of application of the method is to systems with spin-orbit interaction and spin-splitted Fermi surfaces Thank you for listening!